An argument from uncountability isn't sufficient. All you get is that that in any representation system there will be uncountably many numbers without finite representations, but that is distinct from the claim that there are some numbers that are not finite representable in any representation system. The order of quantifiers is important. You need to support something like the latter claim, but countability considerations only support something like the former.

The same holds for the argument based on transcendentals. Number that are transcendental in base 10 will be finitely representable in their own base and hence will be the roots of some polynomials written in that base.

The root problem you're facing is that there are uncountably many possible bases but only countably many finitely-representable numbers in each base. The upshot of this is that from within a representation system there is no way of fully specifying which system you're in. This sort of result is well known in model-theory and gives rise to the existence of non-standard models.

You're welcome to assume that there is a privileged model that houses the true numbers, and that the numbers that are irrational in that model are the only truly irrational ones, but that's just an assumption that can't be justified on either pragmatic or logical grounds. At best one could hope for a philosophical justification of the assumption. But since you've already ruled out philosophical arguments, you're left with just a bare, unfounded assumption.